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Efficient iterative solution of the three-dimensional Helmholtz equation. (English) Zbl 0929.65089
The authors examine two types of preconditioners for the discrete indefinite Helmholtz equation with Sommerfeld-like boundary conditions. The first is derived by discretization of a related continuous operator, the second uses the block Toeplitz approximation to the desired problem. The resulting preconditioning matrices allow the use of fast transform methods (e.g. fast Fourier transform) and differ from the discrete Helmholtz operator by an operator of low rank. Some numerical experiments presented in the paper demonstrate the efficiency of the method when combined with Krylov subspace iteration. The authors show that the performance of restared GMRES with the proposed preconditioners is relatevely insensitive to the discretization mesh size and the wave number, and the algorithms are highly parallelizable. The presented technique is potentially applicable to inhomogeneous media, exterior domain problems and non-Cartesian grids.
MSC:
65N06Finite difference methods (BVP of PDE)
35J05Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation
65F10Iterative methods for linear systems
65F35Matrix norms, conditioning, scaling (numerical linear algebra)